Relaxed Schedulers Can Efficiently Parallelize Iterative Algorithms

There has been significant progress in understanding the parallelism inherent to iterative sequential algorithms: for many classic algorithms, the depth of the dependence structure is now well understood, and scheduling techniques have been developed to exploit this shallow dependence structure for efficient parallel implementations. A related, applied research strand has studied methods by which certain iterative task-based algorithms can be efficiently parallelized via relaxed concurrent priority schedulers. These allow for high concurrency when inserting and removing tasks, at the cost of executing superfluous work due to the relaxed semantics of the scheduler. In this work, we take a step towards unifying these two research directions, by showing that there exists a family of relaxed priority schedulers that can efficiently and deterministically execute classic iterative algorithms such as greedy maximal independent set (MIS) and matching. Our primary result shows that, given a randomized scheduler with an expected relaxation factor of $k$ in terms of the maximum allowed priority inversions on a task, and any graph on $n$ vertices, the scheduler is able to execute greedy MIS with only an additive factor of poly($k$) expected additional iterations compared to an exact (but not scalable) scheduler. This counter-intuitive result demonstrates that the overhead of relaxation when computing MIS is not dependent on the input size or structure of the input graph. Experimental results show that this overhead can be clearly offset by the gain in performance due to the highly scalable scheduler. In sum, we present an efficient method to deterministically parallelize iterative sequential algorithms, with provable runtime guarantees in terms of the number of executed tasks to completion.Comment: PODC 2018, pages 377-386 in proceeding

The Lock-free kk-LSM Relaxed Priority Queue

Priority queues are data structures which store keys in an ordered fashion to allow efficient access to the minimal (maximal) key. Priority queues are essential for many applications, e.g., Dijkstra's single-source shortest path algorithm, branch-and-bound algorithms, and prioritized schedulers. Efficient multiprocessor computing requires implementations of basic data structures that can be used concurrently and scale to large numbers of threads and cores. Lock-free data structures promise superior scalability by avoiding blocking synchronization primitives, but the \emph{delete-min} operation is an inherent scalability bottleneck in concurrent priority queues. Recent work has focused on alleviating this obstacle either by batching operations, or by relaxing the requirements to the \emph{delete-min} operation. We present a new, lock-free priority queue that relaxes the \emph{delete-min} operation so that it is allowed to delete \emph{any} of the $\rho+1$ smallest keys, where $\rho$ is a runtime configurable parameter. Additionally, the behavior is identical to a non-relaxed priority queue for items added and removed by the same thread. The priority queue is built from a logarithmic number of sorted arrays in a way similar to log-structured merge-trees. We experimentally compare our priority queue to recent state-of-the-art lock-free priority queues, both with relaxed and non-relaxed semantics, showing high performance and good scalability of our approach.Comment: Short version as ACM PPoPP'15 poste

Data Structures for Task-based Priority Scheduling

Many task-parallel applications can benefit from attempting to execute tasks in a specific order, as for instance indicated by priorities associated with the tasks. We present three lock-free data structures for priority scheduling with different trade-offs on scalability and ordering guarantees. First we propose a basic extension to work-stealing that provides good scalability, but cannot provide any guarantees for task-ordering in-between threads. Next, we present a centralized priority data structure based on $k$-fifo queues, which provides strong (but still relaxed with regard to a sequential specification) guarantees. The parameter $k$ allows to dynamically configure the trade-off between scalability and the required ordering guarantee. Third, and finally, we combine both data structures into a hybrid, $k$-priority data structure, which provides scalability similar to the work-stealing based approach for larger $k$, while giving strong ordering guarantees for smaller $k$. We argue for using the hybrid data structure as the best compromise for generic, priority-based task-scheduling. We analyze the behavior and trade-offs of our data structures in the context of a simple parallelization of Dijkstra's single-source shortest path algorithm. Our theoretical analysis and simulations show that both the centralized and the hybrid $k$-priority based data structures can give strong guarantees on the useful work performed by the parallel Dijkstra algorithm. We support our results with experimental evidence on an 80-core Intel Xeon system

The Power of Choice in Priority Scheduling

Consider the following random process: we are given $n$ queues, into which elements of increasing labels are inserted uniformly at random. To remove an element, we pick two queues at random, and remove the element of lower label (higher priority) among the two. The cost of a removal is the rank of the label removed, among labels still present in any of the queues, that is, the distance from the optimal choice at each step. Variants of this strategy are prevalent in state-of-the-art concurrent priority queue implementations. Nonetheless, it is not known whether such implementations provide any rank guarantees, even in a sequential model. We answer this question, showing that this strategy provides surprisingly strong guarantees: Although the single-choice process, where we always insert and remove from a single randomly chosen queue, has degrading cost, going to infinity as we increase the number of steps, in the two choice process, the expected rank of a removed element is $O( n )$ while the expected worst-case cost is $O( n \log n )$. These bounds are tight, and hold irrespective of the number of steps for which we run the process. The argument is based on a new technical connection between "heavily loaded" balls-into-bins processes and priority scheduling. Our analytic results inspire a new concurrent priority queue implementation, which improves upon the state of the art in terms of practical performance

Non-blocking Priority Queue based on Skiplists with Relaxed Semantics

Priority queues are data structures that store information in an orderly fashion. They are of tremendous importance because they are an integral part of many applications, like Dijkstra’s shortest path algorithm, MST algorithms, priority schedulers, and so on. Since priority queues by nature have high contention on the delete_min operation, the design of an efficient priority queue should involve an intelligent choice of the data structure as well as relaxation bounds on the data structure. Lock-free data structures provide higher scalability as well as progress guarantee than a lock-based data structure. That is another factor to be considered in the priority queue design. We present a relaxed non-blocking priority queue based on skiplists. We address all the design issues mentioned above in our priority queue. Use of skiplists allows multiple threads to concurrently access different parts of the skiplist quickly, whereas relaxing the priority queue delete_min operation distributes contention over the skiplist instead of just at the front. Furthermore, a non-blocking implementation guarantees that the system will make progress even when some process fails. Our priority queue is internally composed of several priority queues, one for each thread and one shared priority queue common to all threads. Each thread selects the best value from its local priority queue and the shared priority queue and returns the value. In case a thread is unable to delete an item, it tries to spy items from other threads\u27 local priority queues. We experimentally and theoretically show the correctness of our data structure. We also compare the performance of our data structure with other variations like priority queues based on coarse-grained skiplists for both relaxed and non-relaxed semantics